
The \(\mathrm {USp}(2N)\)  theory with the \(\mathrm {Spin}(8)\)  mass deformation is realized by embedding one complex dimension of the M5 brane world volume in a complex surface, \(Q_0\) . \(Q_0\)  carries a hyperkähler structure — from which the SW 1-form is derived — and is a deformation of a \((T^2\times \mathbb {C})/\mathbb {Z}_2\)  orbifold. This surface can be thought of (we will be more precise below) as fibered over \(T^2/\mathbb {Z}_2\) .

The intersection with the M5 brane then gives a curve which projects to an \(N\) -fold cover of \(T^2/\mathbb {Z}_2\)  minus one of the orbifold points. At the missing orbifold point the M5 brane is not transverse to \(Q_0\) ; we will call this point the “pole" of the M5 brane.

The \(\mathrm {SU}(2)\)  mass deformation, \(M\) , is realized by further deforming the background surface to \(Q_M\) . Following the discussion of the analogous deformation of the elliptic model in [1]}, describe \(Q_M\)  by two charts to \(Q_0\) , one including the fibers above a neighborhood of a chosen point \(p \in T^2/\mathbb {Z}_2\) , and the other encompassing the rest of the surface. The two coordinate patches are isomorphic to the corresponding patches of \(Q_0\) , and the \(M\)  deformation is realized by requiring that the transition map is a shift of the fiber coordinate which has a pole with residue proportional to \(M\)  at \(p\) . We call this transition map the “\(M\)  shift". Changing \(p\)  and the form of the transition map but keeping \(M\)  fixed does not change the complex structure of \(Q_M\) .

The M5 brane curve for the mass-deformed \(\mathrm {USp}(2N)\)  \(N_f=4\)  SCFT is then locally a degree-\(N\)  polynomial in the fiber coordinate on \(Q_M\)  whose coefficients have at most a simple pole over a chosen orbifold point of \(T^2/\mathbb {Z}_2\) .

